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Results tagged with mg.metric-geometry
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user 6094
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
8
votes
Accepted
Determining the stretch of a cluster of points
There are many roundness measures that have been explored for different applications, which may give you ideas.
A good source for roundness in image processing is this paper, which analyzes
and compar …
- 151k
3
votes
Accepted
Alexandrov curvature of a compact length space
From "A.D. Alexandrov spaces with curvature bounded below",
Burago, Y. and Gromov, M. and Perel'man, G.,
Russian Mathematical Surveys, 47, 1992, p.5:
A locally complete space $Μ$ with intrinsic me …
- 151k
2
votes
Smallest ball containing the intersection of a family of balls
Even though this doesn't answer any of your questions,
it might help to look at the literature on the Kneser–Poulsen conjecture:
If a finite set of balls is repositioned so that the distance between t …
- 151k
12
votes
Feasibility of a list of prescribed distances in R^3
What did not seem to be mentioned in this old thread
is that this is a well-studied problem that goes under the name
distance geometry, which Wikipedia defines as "the characterization and study of se …
- 151k
7
votes
Advanced view of the napkin ring problem?
This is my attempt to understand David Eppstein's construction.
The green segment is the "stick connecting the center of the sphere to a point halfway from top to bottom on the inner surface of the ho …
- 151k
1
vote
3
votes
Some questions about polyhedra in 3-dimensional Euclidean space, E3
Concerning your last question (conditions for a set of points to be realized as the vertices of a polyhedron), it was established by Branko Grünbaum that every non-coplanar finite set of four or more …
- 151k
6
votes
Accepted
Smallest value of largest angle in finite planar configurations
The key bound is $(1 - 1/n) \pi$, due to Erdős and Szekeres:
The above is an excerpt from this paper:
The Erdős-Szekeres result is in their 1961 paper, "On some extremum pr …
- 151k
1
vote
A question about maximizing the volume of a particular kind of convex set
It seems you could curve the base of the cone, making it a sector of a sphere of
radius $r > d$ slightly larger than $d$, centered at a point $c$ directly above the
apex of the cone:
(Arrows indicate …
- 151k
3
votes
Is there always a maximum anti-rectangle with a corner square?
Could you please clarify your definitions via the example polygon below?
Is $\{a,b,c,1,12\}$ an antirectangle of size $5$?
Is this a maximal antirectangle?
What is a largest antirectangle that inc …
- 151k
4
votes
Accepted
Is there interesting structure in the space of Voronoi diagrams?
Although I am not sure I fully understand the question, perhaps this attempt at a partial
answer will clarify matters.
First, it is known that every tree of at least one edge and with
no vertices of …
- 151k
7
votes
0
answers
250
views
Equiareal shapes in $\mathbb{R}^d$
There was quite a bit of work on the so-called
equichordal problem throughout the 20th century, to decide if some plane convex
curve could have two equichordal points.
A point is equichordal for a clo …
- 151k
3
votes
Accepted
Mahler Volume of the Snub 24-Cell
This is just a comment embellished with an image. It seems that the paper you cite,
in its earlier arXiv version,
Mehmet Koca, Mudhahir Al-Ajmi, Nazife Ozdes Koca.
"Quaternionic Representation o …
- 151k
4
votes
1
answer
787
views
When does the intersection of cylinders produce a ball?
Suppose one intersects unit-radius solid cylinders
in $\mathbb{R}^3$, with each cylinder axis passing through
the origin. For example, two such cylinders produce
the Steinmetz solid.
But if we imagin …
- 151k
2
votes
Peeling a polygonal vegetable
This subverts your intent, but in this example, $P(t)$ cannot remain connected as it must
jump to other hole, and then outside the pulp.
So perhaps you should assume that the pulp polygon …
- 151k